- FindStat

Identifier

St001065: Dyck paths ⟶ ℤ

Values

[1,0] => 2

[1,0,1,0] => 3

[1,1,0,0] => 3

[1,0,1,0,1,0] => 5

[1,0,1,1,0,0] => 4

[1,1,0,0,1,0] => 4

[1,1,0,1,0,0] => 3

[1,1,1,0,0,0] => 4

[1,0,1,0,1,0,1,0] => 7

[1,0,1,0,1,1,0,0] => 6

[1,0,1,1,0,0,1,0] => 5

[1,0,1,1,0,1,0,0] => 5

[1,0,1,1,1,0,0,0] => 5

[1,1,0,0,1,0,1,0] => 6

[1,1,0,0,1,1,0,0] => 5

[1,1,0,1,0,0,1,0] => 5

[1,1,0,1,0,1,0,0] => 6

[1,1,0,1,1,0,0,0] => 4

[1,1,1,0,0,0,1,0] => 5

[1,1,1,0,0,1,0,0] => 4

[1,1,1,0,1,0,0,0] => 3

[1,1,1,1,0,0,0,0] => 5

[1,0,1,0,1,0,1,0,1,0] => 9

[1,0,1,0,1,0,1,1,0,0] => 8

[1,0,1,0,1,1,0,0,1,0] => 7

[1,0,1,0,1,1,0,1,0,0] => 7

[1,0,1,0,1,1,1,0,0,0] => 7

[1,0,1,1,0,0,1,0,1,0] => 7

[1,0,1,1,0,0,1,1,0,0] => 6

[1,0,1,1,0,1,0,0,1,0] => 7

[1,0,1,1,0,1,0,1,0,0] => 8

[1,0,1,1,0,1,1,0,0,0] => 6

[1,0,1,1,1,0,0,0,1,0] => 6

[1,0,1,1,1,0,0,1,0,0] => 5

[1,0,1,1,1,0,1,0,0,0] => 5

[1,0,1,1,1,1,0,0,0,0] => 6

[1,1,0,0,1,0,1,0,1,0] => 8

[1,1,0,0,1,0,1,1,0,0] => 7

[1,1,0,0,1,1,0,0,1,0] => 6

[1,1,0,0,1,1,0,1,0,0] => 6

[1,1,0,0,1,1,1,0,0,0] => 6

[1,1,0,1,0,0,1,0,1,0] => 7

[1,1,0,1,0,0,1,1,0,0] => 6

[1,1,0,1,0,1,0,0,1,0] => 8

[1,1,0,1,0,1,0,1,0,0] => 9

[1,1,0,1,0,1,1,0,0,0] => 7

[1,1,0,1,1,0,0,0,1,0] => 5

[1,1,0,1,1,0,0,1,0,0] => 5

[1,1,0,1,1,0,1,0,0,0] => 6

[1,1,0,1,1,1,0,0,0,0] => 5

[1,1,1,0,0,0,1,0,1,0] => 7

[1,1,1,0,0,0,1,1,0,0] => 6

[1,1,1,0,0,1,0,0,1,0] => 6

[1,1,1,0,0,1,0,1,0,0] => 7

[1,1,1,0,0,1,1,0,0,0] => 5

[1,1,1,0,1,0,0,0,1,0] => 5

[1,1,1,0,1,0,0,1,0,0] => 6

[1,1,1,0,1,0,1,0,0,0] => 6

[1,1,1,0,1,1,0,0,0,0] => 4

[1,1,1,1,0,0,0,0,1,0] => 6

[1,1,1,1,0,0,0,1,0,0] => 5

[1,1,1,1,0,0,1,0,0,0] => 4

[1,1,1,1,0,1,0,0,0,0] => 3

[1,1,1,1,1,0,0,0,0,0] => 6

[1,0,1,0,1,0,1,0,1,0,1,0] => 11

[1,0,1,0,1,0,1,0,1,1,0,0] => 10

[1,0,1,0,1,0,1,1,0,0,1,0] => 9

[1,0,1,0,1,0,1,1,0,1,0,0] => 9

[1,0,1,0,1,0,1,1,1,0,0,0] => 9

[1,0,1,0,1,1,0,0,1,0,1,0] => 9

[1,0,1,0,1,1,0,0,1,1,0,0] => 8

[1,0,1,0,1,1,0,1,0,0,1,0] => 9

[1,0,1,0,1,1,0,1,0,1,0,0] => 10

[1,0,1,0,1,1,0,1,1,0,0,0] => 8

[1,0,1,0,1,1,1,0,0,0,1,0] => 8

[1,0,1,0,1,1,1,0,0,1,0,0] => 7

[1,0,1,0,1,1,1,0,1,0,0,0] => 7

[1,0,1,0,1,1,1,1,0,0,0,0] => 8

[1,0,1,1,0,0,1,0,1,0,1,0] => 9

[1,0,1,1,0,0,1,0,1,1,0,0] => 8

[1,0,1,1,0,0,1,1,0,0,1,0] => 7

[1,0,1,1,0,0,1,1,0,1,0,0] => 7

[1,0,1,1,0,0,1,1,1,0,0,0] => 7

[1,0,1,1,0,1,0,0,1,0,1,0] => 9

[1,0,1,1,0,1,0,0,1,1,0,0] => 8

[1,0,1,1,0,1,0,1,0,0,1,0] => 10

[1,0,1,1,0,1,0,1,0,1,0,0] => 11

[1,0,1,1,0,1,0,1,1,0,0,0] => 9

[1,0,1,1,0,1,1,0,0,0,1,0] => 7

[1,0,1,1,0,1,1,0,0,1,0,0] => 7

[1,0,1,1,0,1,1,0,1,0,0,0] => 8

[1,0,1,1,0,1,1,1,0,0,0,0] => 7

[1,0,1,1,1,0,0,0,1,0,1,0] => 8

[1,0,1,1,1,0,0,0,1,1,0,0] => 7

[1,0,1,1,1,0,0,1,0,0,1,0] => 7

[1,0,1,1,1,0,0,1,0,1,0,0] => 8

[1,0,1,1,1,0,0,1,1,0,0,0] => 6

[1,0,1,1,1,0,1,0,0,0,1,0] => 7

[1,0,1,1,1,0,1,0,0,1,0,0] => 8

[1,0,1,1,1,0,1,0,1,0,0,0] => 8

[1,0,1,1,1,0,1,1,0,0,0,0] => 6

>>> Load all 196 entries. <<<

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Description

Number of indecomposable reflexive modules in the corresponding Nakayama algebra.

References

[1] https://en.wikipedia.org/wiki/Torsionless_module

Code

DeclareOperation("IsNthSyzygy",[IsList]);

InstallMethod(IsNthSyzygy, "for a representation of a quiver", [IsList],0,function(LIST)

local M, n, f, N, i, h,W;

M:=LIST[1];

n:=LIST[2];

N:=DualOfModule(NthSyzygy(DualOfModule(M),n));

W:=NthSyzygy(N,n);

if IsDirectSummand(M,W)=true
then
return(1);
else return(0);
fi;
end);


DeclareOperation("NumberOfReflexiveModules",[IsList]);

InstallMethod(NumberOfReflexiveModules, "for a representation of a quiver", [IsList],0,function(LIST)

local M, n, f, N, i, h,W;

L:=LIST[1];

A:=NakayamaAlgebra(L,GF(3));
L:=ARQuiver([A,1000])[2];
LL1:=Filtered(L,x->DominantDimensionOfModule(x,30)>=1);
LL2:=Filtered(LL1,x->IsNthSyzygy([x,2])=1);
return(Size(LL2));


end);

Created

Dec 30, 2017 at 17:27 by Rene Marczinzik

Updated

Dec 30, 2017 at 17:27 by Rene Marczinzik